
First, our proposed remainder-form decomposition functions apply to a new broader class of nonsmooth nonlinear systems, which we call either-sided locally Lipschitz continuous (ELLC) systems,  that is proven in the paper to be a superset of locally Lipschitz continuous (LLC) systems. This new system class relaxes the (almost everywhere) smoothness and bounded gradient requirements and allows nonsmooth vector fields with only one-sided bounded Clarke Jacobians.

Our construction approach for remainder-form decomposition functions is tractable (i.e., computable in closed-form with a non-iterative algorithm) and proven to be the tightest for this family of decomposition functions, which we also show to include [1]}, thus generalizing and improving on the method in [1]}.

We further show that the over-approximation of the image sets of ELLC systems converges to the true tightest enclosing interval at least linearly, when the interval domain width goes to zero and  moreover, a subdivision principle  applies for improving the enclosure of the range/image set.

We introduce a novel set inversion algorithm based on mixed-monotone decomposition functions as an alternative to SIVIA [3]} and the refinement algorithm in [4]}, which enables the design of algorithms for constrained reachability and interval observers for systems with known constraints, modeling redundancy and/or sensor measurements.

